Transcript 5.1 Estimating with Finite Sums

5.1
Estimating with Finite Sums
Quick Review
1. A train travels at 80 mph for 4 hours. How far does it travel?
2. Beginning at a standstill, a car maintains a constant acceleration
of 5 ft/sec for 10 seconds. What is its velocity after 10 seconds?
3. A pump working at 100 gallons/minute pumps for two hours.
How many gallons are pumped?
4. At 8:00pm, the temperature began dropping at a rate of 1 degree
Celcius per hour. Twelve hours later it began rising at a rate of
1.5 degrees per hour for six hours. What was the net change in
temperature over the 18-hour period?
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Quick Review Solutions
1. A train travels at 80 mph for 4 hours. How far does it travel? 320 miles
2. Beginning at a standstill, a car maintains a constant acceleration
of 5 ft/sec for 10 seconds. What is its velocity after 10 seconds? 50 ft/s
3. A pump working at 100 gallons/minute pumps for two hours.
How many gallons are pumped? 12,000 gallons
4. At 8:00pm, the temperature began dropping at a rate of 1 degree
Celcius per hour. Twelve hours later it began rising at a rate of
1.5 degrees per hour for six hours. What was the net change in
temperature over the 18-hour period? 3
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What you’ll learn about
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Distance Traveled
Rectangular Approximation Method (RAM)
Volume of a Sphere
Cardiac Output
Essential Questions
How does learning about estimating with finite
sums help us understand integral calculus.
Example Finding Distance Traveled
when Velocity Varies
1. A particle starts at x  0 and moves along the x-axis with velocity v(t )  t
for time t  0. Where is the particle at t  3?
Graph v and partition the time interval into subintervals of length t. If you use
t  1/ 4, you will have 12 subintervals. The area of each rectangle approximates
the distance traveled over the subinterval. Adding all of the areas (distances)
gives an approximation to the total area under the curve (total distance traveled)
from t  0 to t  3.
v
d = 300 mi
60
mph
hours
5
t
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Example Finding Distance Traveled
when Velocity Varies
Continuing in this manner, derive the area 1/ 4  m  for each subinterval and
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i
add them:
1
9
25 49
81 121 169 225 289 361 441 529 2300
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256 256 256 256 256 256 256 256 256 256 256 256 256
 8.98
LRAM, MRAM, and RRAM
approximations to the area under the
graph of y=x2 from x=0 to x=3
Example Estimating Area Under the
Graph of a Nonnegative Function
2. Estimate the area under the graph of f ( x)  x sin x from x  0 to x  3.
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Apply the RAM program (found in the Technology Resource Manual that
accompanies this textbook).
Using 1000 subintervals, you find the left endpoint
approximate area of 5.77476
Example Finding Volume
2. Estimate the volume of a sphere with a radius of 4 meters.
Pg. 251, 5.1 #1-35 odd